Robust ellipse positioning method and device based on sparse regularization and ADMM

Abstract

The invention discloses a robust ellipse positioning method and device based on sparse regularization and ADMM. The method comprises the following steps: acquiring known parameters required when a target is positioned by using an ellipse positioning method; the known parameters are substituted into the BR measurement abnormal value number optimization problem; according to the BR measurement abnormal value quantity optimization problem, under the constraint condition that dense inner circumference noise energy is bounded, the sparse regularization optimization problem of the target position and sparse abnormal noise is solved with the purpose of minimizing the sparse abnormal noise quantity; the sparse abnormal noise is a measurement error generated by external environment interference and multipath propagation; and solving the BR measurement abnormal value quantity optimization problem based on an ADMM algorithm, and finally obtaining a target positioning result. According to the method, the influence of the BR measurement abnormal value is effectively inhibited, the robustness and the target positioning precision of the ellipse positioning method are remarkably improved, and the method is more efficient and more suitable for real-time online processing.

Description

Technical Field

[0001] The present invention belongs to the technical field of distributed radar cooperative detection, and particularly relates to a robust ellipse positioning method and device based on sparse regularization and ADMM. Background Art

[0002] Distributed multiple-input multiple-output (MIMO) radar is a radar system with multiple transmitting units and multiple receiving units. Ellipse positioning is one of the most common positioning paradigms for distributed MIMO radar to locate targets. Ellipse positioning relies on bistatic range (BR) measurements to estimate the target position. When using distributed MIMO radar to locate a target, each noiseless BR measurement precisely defines an ellipse in a two-dimensional space where the target is located, and the positions of the transmitter-receiver pairs are the foci of the ellipse. Therefore, the intersection of the ellipses determined by multiple transmitter-receiver pairs is the estimated target position.

[0003] Most of the existing ellipse positioning methods are proposed under the least squares framework and perform excellently in small noise scenarios. However, in the actual complex electromagnetic environment, due to factors such as low signal-to-interference-plus-noise ratio and non-line-of-sight signal propagation, BR measurements are often interfered by abnormal noise. The existence of these outliers will cause a significant decline in the performance of the existing least squares positioning methods. [[ID=...]]

[0004] To address the problem of outlier interference in distributed MIMO radar ellipse positioning, researchers have proposed various solutions. Early research mainly focused on outlier detection and rejection. For example, the threshold detection method based on residual analysis identifies and rejects BR measurements with large deviations by setting an outlier threshold. This method has been applied in the research of existing distributed radar systems and achieved certain results. In addition, there are also ellipse positioning methods that group measurement data through clustering analysis and reject abnormal clusters. However, such explicit outlier detection methods often rely on the setting of thresholds or statistical models, are prone to false alarms or missed detections, and are difficult to fully adapt to complex and changing actual scenarios. Therefore, to overcome the limitations of explicit outlier detection, subsequent research has turned to robust estimation methods for outlier suppression and made certain progress, such as: the method based on the maximum correlation entropy criterion (MCC), which formulates the positioning problem as a semidefinite relaxation problem and utilizes the MCC enhancement algorithm for the robustness against outliers; the method based on the Lagrangian programming neural network (LPNN), which suppresses the influence of outliers by solving the -norm minimization problem, effectively reducing the influence of outliers; the method based on message passing (MP), which uses factor graphs to solve The ℓ0 - norm minimization problem realizes outlier suppression and improves the computational efficiency of the solution; the MM method introduces a hyperparameter to average the influence of outliers and uses the majorization-minimization (MM) framework to solve the non-smooth and non-convex robust localization problem, improving the localization accuracy.

[0005] However, when using a distributed MIMO radar for target localization under the condition of outlier interference, although the existing robust estimation methods for outlier suppression show a certain degree of robustness to outliers, there is still much room for improvement in the robustness and localization accuracy of these methods. Summary of the Invention

[0006] To solve the above problems existing in the prior art, the present invention provides a robust elliptical localization method and device based on sparse regularization and ADMM.

[0007] The technical problems to be solved by the present invention are realized through the following technical solutions:

[0008] In a first aspect, the present invention provides a robust elliptical localization method based on sparse regularization and ADMM, including:

[0009] Obtain the known parameters required for locating a target using the elliptical localization method;

[0010] Substitute the known parameters into the BR measurement outlier number optimization problem; the BR measurement outlier number optimization problem is a sparse regularization optimization problem for solving the target position and sparse outlier noise with the goal of minimizing the number of sparse outlier noises under the constraint that the energy of the dense inner noise is bounded; the sparse outlier noise is the measurement error caused by external environmental interference and multipath propagation;

[0011] Solve the BR measurement outlier number optimization problem based on the ADMM algorithm to obtain the target localization result.

[0012] Optionally, the construction method of the BR measurement outlier number optimization problem includes:

[0013] Construct a BR measurement equation according to the BR measurement data;

[0014] Optimize the BR measurement equation by modeling the BR measurement noise as two parts: sparse outlier noise and dense inner noise;

[0015] Construct the BR measurement outlier number optimization problem according to the optimized BR measurement equation.

[0016] Optionally, the BR measurement outlier number optimization problem is expressed as:

[0017]

[0018] where, ||z||0 represents the number of non-zero elements in the sparse anomaly noise vector z, r represents the BR measurement data, Φ(x) represents the bistatic distance sum, x represents the target position vector, and the constraint ||r - Φ(x) - z||2 ≤ ε indicates that the dense inner noise energy is bounded, where ε is an empirical parameter.

[0019] Optionally, solving the BR measurement outlier number optimization problem based on the ADMM algorithm to obtain the target positioning result includes:

[0020] relaxing the BR measurement outlier number optimization problem into an -norm minimization problem of sparse anomaly noise;

[0021] constructing an unconstrained optimization problem with a penalty function form associated with the -norm minimization problem;

[0022] transforming the unconstrained optimization problem into an equivalent constrained optimization problem;

[0023] constructing an augmented Lagrangian function of the constrained optimization problem;

[0024] According to the augmented Lagrangian function, constructing a first sub-problem, a second sub-problem, and a third sub-problem as ADMM sub-problems; the first sub-problem is used to solve the target position vector under the goal of minimizing the augmented Lagrangian function value, the second sub-problem is used to solve the sparse anomaly noise vector under the goal of minimizing the augmented Lagrangian function value, and the third sub-problem is used to solve the auxiliary variable under the goal of minimizing the augmented Lagrangian function value; the auxiliary variable is u = Φ(x) + z - r;

[0025] By iteratively solving the first sub-problem, the second sub-problem, and the third sub-problem, the solution of the BR measurement outlier number optimization problem is realized. When the iterative termination condition is satisfied, the target positioning result is obtained; where, in each round of iteration, the results of the first sub-problem, the second sub-problem, and the third sub-problem are sequentially solved, and the Lagrange multiplier vector is updated according to the solution results.

[0026] Optionally, the augmented Lagrangian function is:

[0027]

[0028] where is the Lagrange multiplier vector, <·,·> represents the vector inner product, μ is the penalty parameter, μ > 0, u is the auxiliary variable, and ρ is the adjustment parameter.

[0029] Optionally, the iterative termination condition is:

[0030] ‖Φ(xt ) + z t -r - u t ‖2 / ‖u t ‖2 < γ;

[0031] Among them, t represents the index of the outer loop iteration times, and γ is the normalized residual threshold; the outer loop is a loop process that successively solves the results of the first sub - problem, the second sub - problem, and the third sub - problem in each round, and updates the Lagrange multiplier vector according to the solution results until the iteration termination condition is met.

[0032] Optionally, the method further includes:

[0033] During the iterative solution of the first sub - problem, the second sub - problem, and the third sub - problem, the first sub - problem is equivalent to a new first sub - problem. A surrogate function is constructed for the new first sub - problem, and the new first sub - problem is transformed into an optimizable problem by using the surrogate function, so as to solve the BR measurement outlier number optimization problem by iteratively solving this optimizable problem, the second sub - problem, and the third sub - problem;

[0034] The new first sub - problem is:

[0035]

[0036] Among them, t m represents the transmitter position, s l represents the receiver position, is an arbitrary element of a t , a t = z t -r - u t , is an arbitrary element of the Lagrange multiplier vector Λ t , M represents the number of transmitters, and L represents the number of receivers;

[0037] The surrogate function includes a first surrogate function, a second surrogate function, and a third surrogate function;

[0038] The first surrogate function includes: when , at the given point x k , the upper bounds of the ||x - t ||2 term and the ||x - s m ||2 term in the l term of the new first sub - problem; k is the number of inner loop iterations when solving the optimizable problem alone;

[0039] The second surrogate function includes: when , at the given point x kwhere, the upper bounds of the terms of ||x - t in the new first sub - problem constructed by using the Cauchy - Schwarz inequality m ||₂ term and the ||x - s l ||₂ term;

[0040] The third surrogate function is: the upper bound of the ||x - t m ||₂||x - s l ||₂ term in the new first sub - problem constructed based on the basic inequality.

[0041] In a second aspect, the present invention provides a robust ellipse positioning device based on sparse regularization and ADMM, including:

[0042] An acquisition module: used to acquire the known parameters required for positioning a target by using the ellipse positioning method;

[0043] A modeling module: used to substitute the known parameters into the BR measurement outlier number optimization problem; the BR measurement outlier number optimization problem is a sparse regularization optimization problem for solving the target position and sparse outlier noise with the goal of minimizing the number of sparse outlier noises under the constraint that the dense inner - circle noise energy is bounded; the sparse outlier noise is the measurement error caused by external environmental interference and multipath propagation;

[0044] A positioning solution module: used to solve the BR measurement outlier number optimization problem based on the ADMM algorithm to obtain the target positioning result.

[0045] A robust ellipse positioning method provided by the present invention first decomposes the BR measurement noise into dense inner - circle noise and sparse outlier noise, and transforms the robust ellipse positioning problem into a BR measurement outlier number optimization problem through sparse modeling; then decomposes the complex BR measurement outlier number optimization problem into multiple sub - problems that are easy to solve through the ADMM algorithm, estimates the target position by solving the sub - problems and iteratively updating, and finally obtains a robust positioning result under the interference of outliers.

[0046] In addition, when solving the ADMM sub - problem, the present invention optimizes the original objective function by constructing a surrogate function to ensure that the algorithm can converge quickly under limited computing resources. Therefore, the present invention effectively suppresses the influence of BR measurement outliers, significantly improves the robustness of the ellipse positioning method and the target positioning accuracy, and the method is more efficient and more suitable for real - time online processing.

[0047] The following will further elaborate on the present invention in conjunction with the accompanying drawings. Description of the Drawings

[0048] Figure 1It is a schematic flowchart of a robust ellipse positioning method based on sparse regularization and ADMM provided by an embodiment of the present invention;

[0049] Figure 2 It is a schematic diagram of the convergence of a robust ellipse positioning method based on sparse regularization and ADMM provided by an embodiment of the present invention;

[0050] Figure 3 It is a schematic diagram for comparing the positioning accuracy of a robust ellipse positioning method based on sparse regularization and ADMM provided by an embodiment of the present invention with existing methods with respect to different numbers of outliers;

[0051] Figure 4 It is a schematic diagram for comparing the positioning accuracy of a robust ellipse positioning method based on sparse regularization and ADMM provided by an embodiment of the present invention with existing methods with respect to different standard deviations of dense inner noise;

[0052] Figure 5 It is a schematic diagram for comparing the positioning accuracy of a robust ellipse positioning method based on sparse regularization and ADMM provided by an embodiment of the present invention with existing methods as the upper bound of the abnormal noise amplitude changes;

[0053] Figure 6 It is a schematic structural diagram of a robust ellipse positioning device based on sparse regularization and ADMM provided by an embodiment of the present invention. Detailed implementation manners

[0054] The present invention will be further described in detail below with reference to specific embodiments, but the implementation manners of the present invention are not limited thereto.

[0055] In order to effectively suppress the influence of BR measurement outliers and be able to more robustly, accurately and efficiently use the ellipse positioning method to locate the target, an embodiment of the present invention provides a robust ellipse positioning method based on sparse regularization and ADMM. Refer to Figure 1 , and this method includes the following steps:

[0056] S10. Obtain the known parameters required for locating the target using the ellipse positioning method.

[0057] Exemplarily, the known parameters required for locating the target using the ellipse positioning method may include: BR measurement data, transmitter position, receiver position, adjustment parameter ρ, penalty parameter μ, and normalized residual threshold γ. Among them, the BR measurement data is the data calculated in the radar system.

[0058] S20. Substitute the known parameters into the BR measurement outlier number optimization problem.

[0059] Here, the problem of optimizing the number of BR measurement outliers is a sparse regular optimization problem for solving the target position and sparse outlier noise with the goal of minimizing the number of sparse outlier noises under the constraint that the energy of the dense inner - circle noise is bounded. Among them, the dense inner - circle noise is the measurement error caused by the noise of the radar system itself during operation, and the sparse outlier noise is the measurement error caused by external environmental interference and multipath propagation.

[0060] Among them, the construction method of the problem of optimizing the number of BR measurement outliers includes:

[0061] S201. Construct a BR measurement equation according to the BR measurement data.

[0062] Specifically, in a distributed MIMO radar system with d dimensions (d = 2 or 3) consisting of M transmitters and L receivers, the positions of the transmitters and receivers are known. The position of the transmitter is denoted as The position of the receiver is denoted as The target position is the parameter to be estimated.

[0063] For the transmitter - receiver pair {m, l}, when the BR measurement data is known, the constructed BR measurement equation is:

[0064] r m,l = ||x - t m ||2 + ||x - s l ||2 + e m,l ; (1)

[0065] Among them, t m represents the transmitter position, s l represents the receiver position, x represents the target position vector, ||x - t m ||2 represents the distance from the target to the transmitter, ||x - s l ||2 represents the distance from the target to the receiver, e m,l is the BR measurement noise, and r m,l represents the BR measurement data.

[0066] Define φ m,l (x) = ||x - t m ||2 + ||x - s l ||2, then all ML BR measurement equations can be written in vector form:

[0067] r = Φ(x) + e; (2)

[0068] Among them, r represents the BR measurement data, Φ(x) represents the bistatic distance sum, and e represents the BR measurement noise.

[0069] S202. Optimize the BR measurement equation by modeling the BR measurement noise as two parts: sparse outlier noise and dense inlier noise.

[0070] Specifically, if the proportion of BR measurements contaminated by outliers is too high, it is difficult to accurately estimate the target position x. Therefore, assume that only a small number of outliers exist to ensure the estimability of the target position. Thus, decompose the BR measurement noise vector e in the BR measurement equation into two independent components:

[0071] e = z + v; (3)

[0072] where z represents the sparse outlier noise vector, and v represents the dense inlier noise, and the energy of this dense inlier noise is bounded, i.e., ||v||2 ≤ ε.

[0073] From formulas (2) and (3), the optimized BR measurement equation can be obtained:

[0074] r = Φ(x) + z + v. (4)

[0075] S203. Construct an optimization problem for the number of BR measurement outliers according to the optimized BR measurement equation.

[0076] Specifically, on the premise of assuming that the BR measurement outliers are sparse (i.e., the number of outliers is much less than the number of inliers), the robust ellipse localization problem is modeled as a problem of minimizing the number of outliers in formula (4), that is, the optimization problem for the number of BR measurement outliers:

[0077]

[0078] where ||z||0 represents the number of non-zero elements in the sparse outlier noise vector z, and the constraint condition ||r - Φ(x) - z||2 ≤ ε represents that the energy of the dense inlier noise is bounded, and ε is an empirical parameter.

[0079] S30. Solve the optimization problem for the number of BR measurement outliers based on the ADMM algorithm to obtain the target localization result.

[0080] Among them, the ADMM (Alternating Direction Method of Multipliers) algorithm is a method for efficiently solving constrained convex optimization problems, which combines the decomposition ability of the dual ascent method and the convergence advantage of the augmented Lagrangian algorithm. Through the ADMM algorithm, complex problems can be decomposed into sub-problems that are easy to solve, and the complex problems can be solved by alternately optimizing the sub-problems.

[0081] In the present invention, solving the optimization problem for the number of BR measurement outliers based on the ADMM algorithm to obtain the target localization result specifically includes:

[0082] S301. Relax the problem of optimizing the number of BR measurement outliers into a sparse outlier noise -norm minimization problem.

[0083] The problem of optimizing the number of BR measurement outliers obtained in step S20 is a -norm minimization problem, which is a complex combinatorial optimization problem. Moreover, due to the -norm constraint of the non-linear function Φ(x) introducing non-convexity, the difficulty of solving is further increased. Therefore, to simplify the problem, the objective function of the problem of optimizing the number of BR measurement outliers obtained in step S20 -norm is relaxed to its closest convex approximation - -norm, resulting in:

[0084]

[0085] where ||z||1 represents the -norm of the sparse outlier noise vector z.

[0086] S302. Construct an unconstrained optimization problem with a penalty function form associated with the -norm minimization problem.

[0087] Specifically, construct an unconstrained optimization problem with a penalty function form associated with it according to formula (6):

[0088]

[0089] where ρ is a regularization parameter, ρ > 0, z represents the sparse outlier noise vector, r represents the BR measurement data, and Φ(x) represents the bistatic distance sum.

[0090] S303. Transform the unconstrained optimization problem into an equivalent constrained optimization problem.

[0091] Specifically, to apply the ADMM algorithm, introduce an auxiliary variable u = Φ(x) + z - r, and transform the unconstrained optimization problem (7) into an equivalent constrained optimization problem:

[0092]

[0093] where u is the auxiliary variable.

[0094] S304. Construct the augmented Lagrangian function of the constrained optimization problem.

[0095] Here, the augmented Lagrangian function of the constrained optimization problem (8) is:

[0096]

[0097] where, Λ is a Lagrange multiplier vector, <·,·> represents the vector inner product, μ is a penalty parameter, and μ > 0.

[0098] S305. According to the augmented Lagrangian function, construct the first sub-problem, the second sub-problem, and the third sub-problem as ADMM sub-problems. The first sub-problem is used to solve the target position vector under the objective of minimizing the augmented Lagrangian function value, the second sub-problem is used to solve the sparse abnormal noise vector under the objective of minimizing the augmented Lagrangian function value, and the third sub-problem is used to solve the auxiliary variable under the objective of minimizing the augmented Lagrangian function value.

[0099] Specifically, according to the constructed augmented Lagrangian function, the constrained optimization problem (8) can be decomposed into three ADMM sub-problems that are easy to solve, and optimization is achieved by iteratively solving the three ADMM sub-problems. The first sub-problem, the second sub-problem, and the third sub-problem are as follows:

[0100]

[0101] Lagrange multiplier vector:

[0102] Λ t+1 = Λ t + μ(Φ(x t+1 ) + z t+1 - r - u t+1 ); (13)

[0103] where, t is the index of the outer loop iteration times. The outer loop refers to the loop process of sequentially solving the results of the first sub-problem, the second sub-problem, and the third sub-problem in each round, and updating the Lagrange multiplier vector according to the solution results until the iteration termination condition is satisfied.

[0104] S306. By iteratively solving the first sub-problem, the second sub-problem, and the third sub-problem, the solution to the BR measurement outlier number optimization problem is achieved. When the iteration termination condition is satisfied, the target positioning result is obtained. Among them, in each round of iteration, the results of the first sub-problem, the second sub-problem, and the third sub-problem are sequentially solved, and the Lagrange multiplier vector is updated according to the solution results.

[0105] Among them, the first sub-problem is a non-linear least squares estimation problem, and its objective function contains non-smooth terms and non-convex terms, making it difficult to directly optimize. Therefore, the first sub-problem is equivalent to a new first sub-problem, and a surrogate function is constructed for the new first sub-problem, so as to use the surrogate function to transform the new first sub-problem into an optimizable problem, so as to achieve the solution to the BR measurement outlier number optimization problem by iteratively solving the optimizable problem, the second sub-problem, and the third sub-problem.

[0106] Specifically, regarding the specific process of transforming the first sub-problem into an optimizable problem, refer to the following:

[0107] Here, z t -r - u t is a constant. Therefore, define a t = z t -r - u t , and expand the objective function of the first sub-problem to:

[0108]

[0109] where t m is the transmitter location, s l represents the receiver location, is an arbitrary element of a t , is an arbitrary element of the Lagrange multiplier vector Λ t , M represents the number of transmitters, and L represents the number of receivers.

[0110] Expand the square term in formula (14) and ignore the constant term to obtain an equivalent new first sub-problem:

[0111]

[0112] Since the objective function of the new first sub-problem (15) contains non-smooth terms and non-convex terms, making the solution of the new first sub-problem (15) challenging. Therefore, based on the majorization-minimization method, design a smooth and convex surrogate function to replace the non-smooth terms and non-convex terms in the objective function of the new first sub-problem (15), ensuring that the surrogate function tightly upper-bounds the original objective function at the iteration point x k .

[0113] For the term in the new first sub-problem, construct the first surrogate function and the second surrogate function in two cases according to 's sign:

[0114] When , presents non-smoothness. Therefore, at the given point x k , for the term in the new first sub-problem, construct an upper bound for the ||x - t m ||2 term as:

[0115]

[0116] where k is the number of inner loop iterations when solving the optimizable problem separately.

[0117] Similarly, when At a given point x k for the term ||x - s l ||² in the new first sub - problem, construct an upper bound as:

[0118]

[0119] When it shows non - smooth and non - convexity. Therefore, at the given point x using the Cauchy - Schwarz inequality, we get: k (x - t

[0120] (x - t m ) T (x k - t m ) ≤ ||x - t m ||²||x k - t m ||²; (18)

[0121] Dividing both sides of equation (18) by - ||x k - t m ||², for the term ||x - t m ||² in the new first sub - problem, construct an upper bound as:

[0122]

[0123] Similarly, when at a given point x k for the term ||x - s l ||² in the new first sub - problem, construct an upper bound as:

[0124]

[0125] For the non - smooth term ||x - t m ||²||x - s l ||² in the new first sub - problem, based on the basic inequality, that is: b 2 + c 2 - 2bc=(b - c) 2 ≥0, let Construct an upper bound for the ||x - t m ||²||x - s l ||² term in the new first sub - problem as:

[0126]

[0127] Then, substitute the first proxy function (16)(17), the second proxy function (19)(20), and the third proxy function (21) into the new first sub-problem and ignore the constant term to transform the new first sub-problem into an optimizable problem:

[0128]

[0129] where, and respectively represent and the index sets of.

[0130] Then, through the first-order optimality condition, the closed-form solution of the optimizable problem (22) can be derived as:

[0131]

[0132] Thus, based on the idea of the ADMM algorithm, by iteratively solving the optimizable problem, the second sub-problem, and the third sub-problem, the solution to the problem of optimizing the number of BR measurement outliers is achieved. The specific solution process is as follows:

[0133] (a) Iteratively optimize the target position vector x according to Equation (23) until the inner-loop termination condition of the optimizable problem is satisfied:

[0134] ‖x k -x k-1 ‖2 / ‖x k-1 ‖2 < γ;

[0135] where, k is the number of inner-loop iterations when solving the optimizable problem alone, and γ represents the normalized residual threshold.

[0136] After the inner-loop iteration of solving the optimizable problem terminates, take the obtained target position as the solution result of x in the iteration result of the outer loop, denoted as x t+1 .

[0137] (b) Use the subgradient method to solve the second sub-problem.

[0138] The second sub-problem (11) is a convex optimization problem. However, the ‖z‖1 term in the objective function is non-differentiable at z = 0, resulting in the non-smoothness of the objective function. Therefore, the subgradient method is used to find the minimum value of the objective function. From this, the update rule for the sparse outlier noise vector z can be obtained as:

[0139] z i+1 = z i - α i g i ; (24)

[0140] where \(i\) represents the number of inner-loop iterations when solving the second sub-problem alone, and \(\alpha\) i is the step size (fixed or time-varying), and \(g\) i represents the sub-gradient of the objective function.

[0141] \(g\) i can be calculated by the following formula:

[0142] \(g\) i =\(\mu(z\) i +\(\varPhi(x\) t+1 ) - r - u\) t )+\(\varLambda\) t + \(\text{sgn}(z\) i )\); (25)

[0143] where \(\text{sgn}(z\) i ) = [\(\text{sgn}(z_1),...,\text{sgn}(z\) ML )]\) T , and \(\text{sgn}(\cdot)\) represents the sign function.

[0144] Iteratively optimize the sparse anomaly noise vector \(z\) according to Equation (24) until the inner-loop termination condition of the second sub-problem is satisfied:

[0145] \(\|z\) i - z\) i-1 \|_2 / \|z\) i-1 \|_2 < \gamma\);

[0146] where \(i\) represents the number of inner-loop iterations when solving the second sub-problem alone, and \(\gamma\) represents the normalized residual threshold.

[0147] After the inner-loop iteration for solving the second sub-problem terminates, the obtained sparse anomaly noise is used as the solution result of \(z\) in the iteration result of the outer loop, denoted as \(z\) t+1 .

[0148] (c) Solve the third sub-problem.

[0149] The third sub-problem (12) is a linear least squares estimation problem. Therefore, its closed-form solution can be obtained as:

[0150]

[0151] (d) Repeat steps (a)-(c) to determine the target positioning result.

[0152] Specifically, in each outer loop, the target position vector is updated by executing step (a), the sparse anomaly noise vector is updated by executing step (b), the auxiliary variable is updated by executing step (c), and then the Lagrange multiplier vector is updated according to formula (13). The above update process of the outer loop is repeated until the iteration termination condition of the outer loop is satisfied, and the final target positioning result is obtained. Here, the iteration termination condition of the outer loop is:

[0153] ‖Φ(x t )+z t -r-u t ‖2 / ‖u t ‖2<γ;

[0154] where t represents the index of the outer loop iteration times, and γ is the normalized residual threshold.

[0155] Exemplarily, at the beginning of the outer loop, the initial target position x 0 can be any position within the detection range, and the initial sparse anomaly noise vector z 0 , the initial auxiliary vector u 0 and the initial Lagrange multiplier vector Λ 0 can be taken as the zero vector.

[0156] A robust ellipse positioning method based on sparse regularization and ADMM provided by the present invention first decomposes the BR measurement noise into dense inner noise and sparse anomaly noise, and transforms the robust ellipse positioning problem into an optimization problem of the number of BR measurement outliers through sparse modeling; then decomposes the complex BR measurement outlier number optimization problem into multiple sub-problems that are easy to solve through the ADMM algorithm, estimates the target position by solving the sub-problems and iteratively updating, and finally obtains a robust positioning result under the interference of outliers.

[0157] In addition, when solving the ADMM sub-problems, the present invention optimizes the original objective function by constructing a surrogate function to ensure that the algorithm can converge quickly with limited computing resources. Therefore, the present invention effectively suppresses the influence of BR measurement outliers, significantly improves the robustness and target positioning accuracy of the ellipse positioning method, and the method is more efficient and more suitable for real-time online processing.

[0158] The following further illustrates the robust ellipse positioning method based on sparse regularization and ADMM provided by the present invention through simulation experiments.

[0159] (1) Simulation scenario setting.

[0160] Consider a distributed MIMO radar system consisting of M = 6 transmitters and L = 5 receivers. The transmitters and receivers are uniformly distributed on arcs centered at the origin with radii of 300 meters and 500 meters, respectively. The true position of the target is (300, -420) meters. The dense inlier noise measured by BR follows a Gaussian distribution while the outlier noise follows a uniform distribution U(-ζ, ζ) m. To evaluate the positioning accuracy, 2000 Monte Carlo trials are conducted under each test condition to calculate the root mean square error (RMSE).

[0161] Obtain the initial parameters and known parameters. The initial target position is randomly sampled from a uniform distribution U(-300, 300) m. The sparse outlier noise vector z, the auxiliary variable u, and the Lagrange multiplier vector Λ are all initialized as zero vectors. The values of the adjustment parameter ρ and the penalty parameter μ depend on the weights of the penalty term ρ||z||1 in Equation (9) and Here, both ρ and μ are set to 1.

[0162] (2) Simulation results.

[0163] First, study the convergence of the robust elliptical positioning method based on sparse regularization and ADMM proposed in the present invention (hereinafter referred to as the ADMM method). Figure 2 Shows the variation of the normalized residual with the increase in the number of iterations in 500 Monte Carlo trials. It can be seen that the ADMM method converges extremely fast and usually converges within 20 iterations.

[0164] Next, in the scenario where the number of BR measurement outliers varies from 2 to 10, and the standard deviation σ of the inlier noise in and the upper bound ζ of the outlier amplitude are fixed at 10 meters and 100 meters, respectively, test the performance of the proposed ADMM method, MM method, and MP method. Figure 3 Shows the root mean square error (RMSE) of the three methods varying with the number of outliers. It can be seen that the performance of all methods decreases with the increase in the number of outliers, but the proposed ADMM method is significantly better than the MM method and the MP method. The MP method performs the worst and its performance deteriorates sharply when the number of outliers exceeds 4.

[0165] Subsequently, in the scenario where the number of outliers is fixed at 2, ζ remains unchanged, and the standard deviation σ of the inlier noise in varies from 10 meters to 50 meters, test the performance of the proposed ADMM method, MM method, and MP method. Figure 4 Shows the root mean square error of the three methods varying with the standard deviation of the inlier Gaussian noise. It can be seen that the proposed ADMM method still maintains the best performance, while the MP method performs the worst.

[0166] Finally, the number of outliers and the standard deviation of the inner noise are fixed at 2 and 10 meters respectively, and the performance of the proposed ADMM method, MM method and MP method in the presence of large outliers is tested by increasing the value of ζ. Figure 5 The root mean square error of the three methods versus the upper bound ζ of the outlier amplitude is shown. At all five ζ values, the positioning accuracy of the proposed ADMM method is higher than that of the MM method and the MP method. The MP method is only better than the MM method when ζ = 100 m, which may be because the -norm estimator used in the MP method is more robust to large outliers than the -norm estimator with a balance parameter in the MM method.

[0167] The computational complexities of the proposed ADMM method, MM method and MP method are all linearly related to the ML. The following table shows the average computational times measured using MATLAB 2022b on a Windows desktop with an Intel Core i7-13700K 3.40 GHz and 16 GB of RAM. It can be seen that the computational complexity of the proposed ADMM method is comparable to that of the MM method but higher than that of the MP method.

[0168] Method Average calculation time ADMM method 0.0113 MM 0.0367 MP 0.0028

[0169] A robust ellipse positioning method based on sparse regularization and ADMM provided by the present invention first decomposes the BR measurement noise into dense inner noise and sparse outlier noise, and transforms the robust ellipse positioning problem into a BR measurement outlier number optimization problem through sparse modeling; then, the complex BR measurement outlier number optimization problem is decomposed into multiple sub-problems that are easy to solve by the ADMM algorithm, and the target position is estimated by solving the sub-problems and iteratively updating, and finally a robust positioning result is obtained under the interference of outliers.

[0170] In addition, when solving the ADMM sub-problem, the present invention optimizes the original objective function by constructing a surrogate function to ensure that the algorithm can converge quickly under limited computing resources. Therefore, the present invention effectively suppresses the influence of BR measurement outliers, significantly improves the robustness of the ellipse positioning method and the target positioning accuracy, and the method is more efficient and more suitable for real-time online processing.

[0171] Corresponding to the above-mentioned robust ellipse positioning method based on sparse regularization and ADMM, an embodiment of the present invention further provides a robust ellipse positioning device based on sparse regularization and ADMM; as Figure 6 shown, the device may include:

[0172] An acquisition module 601: configured to acquire known parameters required for positioning a target using an ellipse positioning method;

[0173] Modeling module 602: It is used to substitute known parameters into the optimization problem of the number of BR measurement outliers; the optimization problem of the number of BR measurement outliers is a sparse regular optimization problem that aims to minimize the number of sparse abnormal noises under the constraint that the energy of the dense inner noise is bounded, and solve for the target position and the sparse abnormal noises; the sparse abnormal noises are measurement errors caused by external environmental interference and multipath propagation.

[0174] Solution positioning module 603: It is used to solve the optimization problem of the number of BR measurement outliers based on the ADMM algorithm to obtain the target positioning result.

[0175] A robust ellipse positioning device based on sparse regularization and ADMM provided by the present invention first decomposes the BR measurement noise into dense inner noise and sparse abnormal noise, and transforms the robust ellipse positioning problem into an optimization problem of the number of BR measurement outliers through sparse modeling; then, the complex optimization problem of the number of BR measurement outliers is decomposed into multiple sub-problems that are easy to solve through the ADMM algorithm, and the target position is estimated by solving the sub-problems and iteratively updating, and finally a robust positioning result is obtained under the interference of outliers.

[0176] In addition, when solving the ADMM sub-problem in the present invention, the original objective function is optimized by constructing a surrogate function to ensure that the algorithm can converge quickly under limited computing resources. Therefore, the present invention effectively suppresses the influence of BR measurement outliers, significantly improves the robustness of the ellipse positioning method and the target positioning accuracy, and the method is more efficient and more suitable for real-time online processing.

[0177] It should be noted that for the device, since it is basically similar to the method embodiment, the description is relatively simple, and for the relevant parts, please refer to the partial description of the method embodiment.

[0178] It should be noted that the terms "first", "second", etc. are used to distinguish similar objects and do not necessarily have to be used to describe a specific order or sequence. It should be understood that the data used in this way can be interchanged under appropriate circumstances so that the embodiments of the present invention described here can be implemented in an order other than those illustrated or described here. The embodiments described in the following exemplary embodiments do not represent all embodiments consistent with the present invention. On the contrary, they are only examples of devices and methods consistent with some aspects of the present invention.

[0179] In the description of this specification, the description with reference to terms such as "one embodiment", "some embodiments", "examples", "specific examples", or "some examples" means that the specific features or characteristics described in connection with the embodiment or example are included in at least one embodiment or example of the present invention. In this specification, the schematic representations of the above terms do not necessarily refer to the same embodiment or example. Moreover, the specific features or characteristics described can be combined in any one or more embodiments or examples in a suitable manner. In addition, those skilled in the art can combine and combine the different embodiments or examples described in this specification.

[0180] Although the present invention has been described in connection with various embodiments herein, however, in the process of implementing the claimed invention, those skilled in the art can understand and achieve other variations of the disclosed embodiments by viewing the accompanying drawings and the disclosure. In the description of the present invention, the term "comprising" does not exclude other components or steps, the word "a" or "an" does not exclude a plurality of cases, and the meaning of "a plurality" is two or more, unless otherwise specifically defined. In addition, certain measures are described in different embodiments, but this does not mean that these measures cannot be combined to produce good results.

[0181] The above content is a further detailed description of the present invention in combination with specific preferred embodiments, and it cannot be determined that the specific implementation of the present invention is only limited to these descriptions. For those of ordinary skill in the technical field to which the present invention pertains, without departing from the concept of the present invention, several simple deductions or substitutions can be made, and all should be regarded as belonging to the protection scope of the present invention.

Claims

1. A robust elliptical positioning method based on sparse regularization and ADMM, characterized in that Including: Obtaining the known parameters required for positioning a target using the elliptical positioning method; Substituting the known parameters into the BR measurement outlier number optimization problem; the BR measurement outlier number optimization problem is a sparse regular optimization problem for solving the target position and sparse outlier noise with the goal of minimizing the number of sparse outlier noises under the constraint that the energy of the dense inner - circle noise is bounded; The sparse outlier noise is the measurement error caused by external environmental interference and multipath propagation; Solving the BR measurement outlier number optimization problem based on the ADMM algorithm to obtain the target positioning result.

2. The robust ellipse positioning method based on sparse regularization and ADMM according to claim 1, wherein The construction method of the BR measurement outlier number optimization problem includes: Constructing a BR measurement equation according to the BR measurement data; Optimizing the BR measurement equation by modeling the BR measurement noise as two parts: sparse outlier noise and dense inner - circle noise; Constructing the BR measurement outlier number optimization problem according to the optimized BR measurement equation.

3. The robust elliptical positioning method based on sparse regularization and ADMM according to claim 1, wherein The BR measurement outlier number optimization problem is expressed as: where, ||z||0 represents the number of non - zero elements in the sparse outlier noise vector z, r represents the BR measurement data, Φ(x) represents the bistatic distance sum, x represents the target position vector, and the constraint condition ||r - Φ(x)-z||2≤ε represents that the energy of the dense inner - circle noise is bounded, and ε is an empirical parameter.

4. The robust elliptical positioning method based on sparse regularization and ADMM according to claim 3, wherein The solving of the BR measurement outlier number optimization problem based on the ADMM algorithm to obtain the target positioning result includes: Relaxing the BR measurement outlier number optimization problem into an l1 - norm minimization problem of the sparse outlier noise; Constructing an unconstrained optimization problem associated with the l1 - norm minimization problem in the form of a penalty function; Converting the unconstrained optimization problem into an equivalent constrained optimization problem; Constructing an augmented Lagrangian function of the constrained optimization problem; Constructing a first sub - problem, a second sub - problem, and a third sub - problem as ADMM sub - problems according to the augmented Lagrangian function; the first sub - problem is used to solve the target position vector with the goal of minimizing the value of the augmented Lagrangian function, the second sub - problem is used to solve the sparse outlier noise vector with the goal of minimizing the value of the augmented Lagrangian function, and the third sub - problem is used to solve the auxiliary variable with the goal of minimizing the value of the augmented Lagrangian function; the auxiliary variable is u = Φ(x)+z - r; By iteratively solving the first sub - problem, the second sub - problem, and the third sub - problem, the solution of the BR measurement outlier number optimization problem is realized. When the iteration termination condition is satisfied, the target positioning result is obtained; where, in each round of iteration, the results of the first sub - problem, the second sub - problem, and the third sub - problem are sequentially solved, and the Lagrange multiplier vector is updated according to the solution results.

5. The robust elliptical positioning method based on sparse regularization and ADMM according to claim 4, wherein The augmented Lagrangian function is: wherein, is a Lagrange multiplier vector, <·,·> represents the vector inner product, μ is a penalty parameter, μ > 0, u is an auxiliary variable, and ρ is a regulation parameter.

6. The robust elliptical positioning method based on sparse regularization and ADMM according to claim 4, characterized in that The iteration termination condition is: ‖Φ(x t )+z t -r-u t ‖2 / ‖u t ‖2<γ; where, t represents the index of the outer - loop iteration times, γ is the normalized residual threshold; the outer - loop is a loop process that sequentially solves the results of the first sub - problem, the second sub - problem, and the third sub - problem in each round, and updates the Lagrange multiplier vector according to the solution results until the iteration termination condition is satisfied.

7. The robust elliptical positioning method based on sparse regularization and ADMM according to claim 4, wherein The method further includes: During the iterative solution of the first sub-problem, the second sub-problem, and the third sub-problem, the first sub-problem is equivalent to a new first sub-problem. For the new first sub-problem, a surrogate function is constructed, and the new first sub-problem is transformed into an optimizable problem by using the surrogate function. By iteratively solving the optimizable problem, the second sub-problem, and the third sub-problem, the solution to the optimization problem of the number of BR measurement outliers is achieved; The new first sub-problem is: where t m represents the transmitter position, s l represents the receiver position, is an arbitrary element of a t , a t = z t - r - u t , is an arbitrary element of the Lagrange multiplier vector Λ t , M represents the number of transmitters, and L represents the number of receivers; The surrogate function includes a first surrogate function, a second surrogate function, and a third surrogate function; The first proxy function includes: when at a given point x k in the new first sub-problem, the in the term ||x - t m ||2 term and the upper bound of the ||x - s l ||2 term; k is the number of inner loop iterations when solving the optimizable problem separately; The second proxy function includes: when at a given point x k the upper bounds of the ||x - t term and the ||x - s m ||2 term in the l ||2 term in the new first subproblem constructed using the Cauchy - Schwarz inequality. The third proxy function is: the upper bound of the two terms of ||x - t|| m m ||2||x - s l ||2 constructed based on the basic inequality in the new first subproblem.

8. A robust elliptical positioning device based on sparse regularization and ADMM, characterized in that It includes: An acquisition module: used to acquire the known parameters required for positioning a target using the ellipse positioning method; A modeling module: used to substitute the known parameters into the optimization problem of the number of BR measurement outliers; the optimization problem of the number of BR measurement outliers is a sparse regular optimization problem that aims to minimize the number of sparse abnormal noises under the constraint that the energy of the dense inner noise is bounded, and solve for the target position and the sparse abnormal noises; The sparse abnormal noises are measurement errors caused by external environmental interference and multipath propagation; A positioning solution module: used to solve the optimization problem of the number of BR measurement outliers based on the ADMM algorithm to obtain the target positioning result.

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